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Prime number theorem for regular Toeplitz subshifts

Part of: Multiplicative number theory Topological dynamics

Published online by Cambridge University Press:  15 February 2021

KRZYSZTOF FRĄCZEK Open the ORCID record for KRZYSZTOF FRĄCZEK [Opens in a new window] ,
ADAM KANIGOWSKI  and
MARIUSZ LEMAŃCZYK
KRZYSZTOF FRĄCZEK*
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100Toruń, Poland
ADAM KANIGOWSKI
Affiliation:
Department of Mathematics, University of Maryland, 4176 Campus Drive, William E. Kirwan Hall, College Park, MD20742-4015, USA (e-mail: akanigow@umd.edu)
MARIUSZ LEMAŃCZYK
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100Toruń, Poland (e-mail: mlem@mat.umk.pl)
*
e-mail: fraczek@mat.umk.pl
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Abstract

We prove that neither a prime nor an l-almost prime number theorem holds in the class of regular Toeplitz subshifts. But when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler’s totient function is assumed, then the two theorems hold.

Keywords

almost prime numberspolynomial ergodic theoremsprime number theoremToeplitz systems

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 11N05: Distribution of primes 11N13: Primes in progressions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 4 , April 2022 , pp. 1446 - 1473
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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